Question

A particle of sass 20g is released with an initial velocity $5{\text{ m/s}}$ along the curve from point A, as shown in the figure. The point A is at length h from B. The particle slides along the frictionless surface. When the particle reaches point B, its angular momentum O will be: (Take g=$10{\text{ m}}{{\text{s}}^{ - 2}}$)

A) $2{\text{ kg - }}{{\text{m}}^2}{{\text{s}}^{ - 1}}$
B) $8{\text{ kg - }}{{\text{m}}^2}{{\text{s}}^{ - 1}}$
C) $3{\text{ kg - }}{{\text{m}}^2}{{\text{s}}^{ - 1}}$
D) $6{\text{ kg - }}{{\text{m}}^2}{{\text{s}}^{ - 1}}$

Answer 1

In this question, we need to determine the angular momentum of particle B when it reaches at point B. For this we will use the relation as $\vec L = \vec r \times \vec p$, where
$\vec L$ is the angular momentum of a moving particle about a point
$\vec r$ is the length of perpendicular on the line of motion
$\vec p$ is the component of momentum along the perpendicular to r

Complete step by step answer:
Mass of the particle $m = 20g$
The velocity of the particle at point a is ${V_a} = 5{\text{ m/s}}$
Since the velocity of the particle will be different for two points, hence we apply work-energy theorem from point A to B
${W_f} = \vartriangle KE - - (i)$
For two points equation (i) can be written as
$mgh = \dfrac{1}{2}mv_B^2 - \dfrac{1}{2}mv_A^2 - - (ii)$
Equation (ii) can be further written as

$$mgh = \dfrac{1}{2}m\left( {v_B^2 - v_A^2} \right) \\\ 2gh = v_B^2 - v_A^2 \\\$$

Now by substitute the values of ${v_A}$and$h$, we get

$$2gh = v_B^2 - v_A^2 \\\ \Rightarrow 2 \times 10 \times 10 = v_B^2 - {\left( 5 \right)^2} \\\ \Rightarrow v_B^2 - 25 = 200 \\\ \Rightarrow v_B^2 = 225 \\\ \Rightarrow {v_B} = 15{\text{ m/s}} \\\$$

Hence the angular momentum about O will be

$$L = mv{r_{OB}} \\\ = \dfrac{{15}}{{1000}} \times 15 \times 20 \\\ = 6{\text{ kg - }}{{\text{m}}^2}{{\text{s}}^{ - 1}} \\\$$

Option D is correct.

Note: Angular momentum is the same as linear momentum, but it is in rotational form. Angular momentum is a vector quantity, and it is the product of a body's rotational inertia and rotational velocity about a particular axis.